Grate lemon peels and place lemon zest in a small bowl. Juice the lemons into a separate small bowl. Whisk the oil, lemon juice, cheese, salt and pepper in a large bowl and blend. Set the lemon sauce aside.

Meanwhile, bring a large pot of salted water to a boil. Add the spaghetti and cook, stirring occasionally until tender but still firm to the bite, about 8 minutes. Drain, reserving 1 cup of the cooking liquid. Add the spaghetti to the lemon sauce, and toss with the basil and lemon zest. Toss the pasta with enough reserved cooking liquid, 1/4 cup at a time, to moisten. Season the pasta with more salt and pepper to taste. Transfer to bowls and serve.